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UNIFORM ESTIMATE ON FINITE TIME RUIN PROBABILITIES WITH RANDOM INTEREST RATE

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UNIFORM ESTIMATE ON FINITE TIME RUIN PROBABILITIES WITH RANDOM INTEREST RATEON,on,On,TIME,RUIN,WITH,RATE,rate,time,ruin

    UNIFORM ESTIMATE ON FINITE TIME RUIN PROBABILITIES WITH RANDOM

    INTEREST RATE

    AvailableonlineatWWW.sciencedirect.com

    

    ScienceDirect

    ActaMathematicaScientia2010,30B(3):688700

    数学物理

    http://actams.wipm.aC.ca

    UNIF0RMESTIMTE0NFINITETIMERUIN

    P_R0BABILITIESWITHRAND0MINTERESTRATE

    MingRuixing(明瑞星)

    SchoolofMathematicsandInformationSciences,~angxiNormalUniversity,Nanchang330022,China

    SchoolofMathematicsandStatistics,WuhanUniversity,Wuhan430072,China HeXiaoxia(何晓霞)

    CollegeolScience,WuhanUniversityolScienceandTechnology,Wuhan430081,China HuYijunf胡亦钧)

    SchoolofMathematicsandStatistics,WuhanUniversity,Wuhan430072,China LiuJuan(刘娟)

    CollegeofMathematicsandStatistics,HubeiNormalUniversity,Huangshi435002China AbstractWleconsideradiscretetimeriskmode1inwhichthenetpayout(insurance risk)f,k=1,2,...areassumedtotakerea]valuesandbelongtotheheavytailed

    classcnandthediscountfactors(financialrisk){,k=1,2,}concentrateon

    0,L],where0<0<1,L<..,{,k=1,2,},and{,k=1,2,}areassumedto

    bemutuallyindependent.Weinvestigatetheasymptoticbehavioroftheruinprobability

    withinafinitetimehorizonastheinitialcapitaltendstoinfinity)andfigureoutthat theconvergenceholdsuniformlyforalln1,whichisdifferentfromTangQHand TsitsiashviliG(AdvApplProb,2004,36:12781299).

    KeywordsRandominterestrate;finitetimeruinprobability;uniformity 2000MRSubjectClassification60K99;60F99

    1Introduction

    Inthisarticle.weconsideradiscretetimeriskmodelwithrandominterestrate.The basicassumptionsofthismodelareasfollows,asappliedbyNyrhinen[9,10]andTangand Tsitsiashvili[12].

    H1.ThesuccessivenetincomesAn,n=1,2,?,constituteasequenceofi.i.d.r.V.'

    withcommond.f.concentratedon(..,..),wherethenetincomeA"isunderstoodasthe

    totalincomingpremiumminusthetotalclaimamountwithinyearn.Thus,wemaysaythat An,n=1,2,,areindependentreplicatesofagenericr.V..

    ReceivedDecember19,2006;revisedOctober7,2008.PartiallysupportedbytheNationalNaturalScience

    FoundationofChina(10671149),theMinistryofEducationofChina,theNaturalScienceFoundationofJiangxi

    (2008GQS0035),andtheFoundationoftheHubeiProvincialDepartmentofEducation(B20091107)

    No.3Mingetal:UNIFORMESTIMATEONFINITETIMERUINPROBABILITIES689 H2.Thereserveiscurrentlyinvestedintoariskyassetwhichmayearnnegativeinterest ratyearn,andr,n=1,2,,alsoconstituteasequenceofi.i.d.r.v.'swithcommond.f concentratedon(1,?).

    H3.Thetwosequences{%,n=1,2,)and{7'n,n=1,2,)aremutuallyindependent.

    Intheliterature,ther.v.Bn=l+rnisoftencalledtheinflationcoefficientfromyearn1

    toyearnandther.V.Yn=Bgthediscountfactorfromyearntoyearnl,=1,2,,

    withgenericr-v.y.IntheterminologyofNorberg[8]jwecallther_v_'sX=AandY

    astheinsuranceriskandfinancialrisk,respectively.Clearly,P(0<y<?)=1.Denote

    Xn=A,=1,2,?--.

    AssumethattheinitialcapitalofaninsurancecompanyisX0.

Weassumethatthenet

    incomeAnismadeorcalculatedattheendoftheyear(=1,2,).Hence,thesurplusof

    thecompanyaccumulatedtilltheendofyearncanbecharacterizedby,whichsatisfiesthe recurrenceequation

    X,Sn=BSn1+A,n

    RelateddiscussionscanbefoundinCai[2andTangandTsitsiashvili[12].Bytherecurrence equation(1.1),weimmediatelyobtain

    n

    ,=

    ?+?n,n=l2,

    j=li=1j=i+l

    n

    where=1byconvention.Thediscountedvalueofsurplusis

    j=n+l

    几几z

    =,=

    II=一?五II,=1,2

    j=li=1j=l

    Thetimeofruinintheconsideredriskmodelwithinitialcapital0isdefinedby 7.()inf{n=1,2,???:<01So}=inf{n1,2,:Sn<01So=)

    Hence,theprobabilityofruinwithinfinitetime,(,n),canbedefinedby (,n)=P(()n)

    andtheultimateruinprobability,(),canbedefinedby

    ()=(,O0)=P(r(x)<?)

    Theruinprobabilityinfinitetimehorizon,(,n),andtheultimateruinprobability,(), areinvestigatedinnumerousliteratures.Nyrhinen[9]junderageneralassumptionthatboth

    {,=1,2,)and{,n=1,2,)aremutuallyindependent,employedlargedeviations techniquesinthediscretetimemodel(1.1)anddeterminedacrudeestimatefortheultimate

    ruinprobability()intheform

    .

im(1og)log()=W

    }OO

    690ACTAMATHEMATICASCIENTIAVb1.3OSet.B

    whereisapositiveconstant.Fortheparticularcasewhereboth{,n=12?')and

    {,:12..)aresequencesofiidr.v.'s,combinedwithTheorem6.3ofGoldie[5]1(?2) impliesastrongerrelationfor(),thatis,

    limx-W()

    Zo.

    =C

    f0rsomeconstantC.TangandTsitsiashvili[12Jconsideredthefinitetimeruinprobability

    (,n)withboth{,n=1,2,???)and{,n=1,2,.??)aresequences.fi?i?d?r?V?'s_They

    provedthatforeachn=1,2,',

    ,

    n)Pf>k=l\i=i

    holdsundertheconditionsthatF?Cn

    P>(seeSection2fordefinition),that

    lim

    )

    fseeSection2fordefinition)andEY<..forsome is,foreach=1,2,.,

    (,n)

    n

    ?P(X>z):=1i=1

    

    10

    NGeta1[7c.nsideredtheparticularcase,whereF?冗一seeSecti.n2fordefinition) and:(1+r)1,i=1,2,...,n,wherer>0isaconstant.Itisobtainedthat,foreach

    n=I,2,?,

    (,))1(1+r)

(1+r)1F(x)

    holds.

    Ingenera1,theuniformityaboutloftenconsiderablymeritsthetheoreticalvalueof theasymptoticrelati.nsobtained.JiangandTang6]extendedtheresults.fNGetal7]t.

    thecasewhereF?ERV(,)seeSection2fordefinition).Itisobtainedthat (,)+r))

    hoidsuniformlyforn1.Hereandhereafter,theuniformityisunderstoodaS limsup

    zo.n>1

    1l=0

    WeiandHu[14extendedtheresult.fJiangandTang[6]t.thecase.f=(1+),i=

    l,2,...,wheren>0,i=1,2,?,areconstants.Theyshowedthat

    holdsuniformly

    F?ERV(-a,

    then,

    (,)一?((1+r)(1+r2)(1+))

    =1

    fOrn1.TangandTsitsiashvili13]provedthatundertheconditionsthat

    ),where0<Q<..,andE(max(y,Y+))<1,forsome0<5<Ol,

    (,fn)n,k?P(?>

    :1\t=1)

    ?

    No.3Mingetal:UNIFORMESTIMATEONFINITETIMERUINPROBABILITIES691 holdsuniformlyforn1.Allowingthediscountfactors'sbeingdependent,Wangetal[15

    showed,undertheconditionF?C(seeSection2fordefinition)togetherwithsomeothermild assumptions,thatrelation(1.10)stillholdsuniformlyforn1.Notethat ERCCCCn

    (seeSection2).Hence,Wangetal[15]generalizedthecorrespondingresultsofTangand

    Tsitsiashvilila].

Inthisarticle,wewillshowthatwhenF?Cnandthediscountfactors'sarei.i.d.

    and?[0,L]fori=1,2,,where0<0<1,L<?,therelation(1.10)stillholdsuniformly

    forn1.Consequently,theresultobtainedherepartiallystrengthenes(1.4)obtainedbyTang andTsitsiashvili12

    bymeanofuniformity,andpartiallygeneralizesthecorrespondingresult ofTangandTsitsiashviliI131fromF?ERVtoF?Cn,ononehand.0ntheotherhand,

    byrestrictingourselvestothecaseofi.i.d.discountfactors's,wealsopartiallyextendthe correspondingresultofWangetal15]fromF?ctoF?cn.

    Therestofthisarticleisorganizedasfollows.InSection2.notationsandmainresultwill beintroduced.InSection3,theproofofthemainresultwillbegiven.

    2NotationsandMainResult

    Throughoutthearticle,weconsiderthedistributionFofrandomvariableXvaluedon (一?,?),thedistributionfunction(d.f.)F(x)=P(x)andF(x)=1F(x)=P(>x)

    isthecorrespondingtaildistributionfunction.LetthedistributionGofrandomvariable Yconcentrateon[0,,where0<0<1,<...Forconvenience,wewriteF1(x)

    kF2(x)andFI(X)kF2(x)(FI(x)=o(F2(x))fork=0)whenlimF1(x)/F2(x)=and limsupF1(x)/Y2(x)k,respectively.F1(x)kF2(x)meanslirainfF1(x)/F~(x)k.If Fx(x)kF2(x)and?0,oo,wealsodenoteitbyF1()F2().

    Inappliedprobabilityandrisktheory,werestrictourinteresttothecaseofheavytails.

    Wemaysimplylistsomeimportantheavy-tailedclassesbelow,whicharerelatedtothepresent article.

    Definition2.1Ad.f.Fwithsupport(一?,?)issaidtobeinERV(,)(Extend

    RegularVariation)forsome0o/<ooifandonlyif

    <li

    

    minr<li

    

    msupF(xy).

    holdsforanyY>1,orequivalently

    li

    minf<li

    

    mSU

    F(xpF(x一—o.1zo.1

    holdsforany0<Y1.WewriteF?ERV(ol,).When=,wesayF?7=己一a(Regular Variation).

    Remark2.1IfXistherandomvariableofthed.fF.wealsosaythatbelongsto

    ),andwritethat?ERV(oz,).Thesimilarsymbolsapplytotheother ERV(-a,

    heavy-tailedclasses.

    692ACTAMATHEMATICA

    SCIENTIAVo1

    .

    30Ser.B

    :

    Denn=I=.n2'2AdFwith8uppOrt()..,..)issaidtObein(DOminatedVariati.n1fiandonlyif,一…)." ?

    

    mSllp<..

    holdsforany0<Y<Iorequivalentlyfo rs0me0<,<1)

    :

    Denn.n2?3Ad?

    f.Fwith8upp.rt()..,..)issait.beinc(L.ng_tajledv_ariati.n)

    fiandonlyif,.………'.""/

    37

    n

    --

    m

400F=f1'

    holdsforanyfixedY?.

    :

    Denn.n2?4Ad? ?Fwithsupp.rt(一?,..)i8saidt.bein(c.nsistentv_ariati.n1

    fiandonlyif…………)"

    1,orequivalently,liraliminf: lzo.Ff)

    ;f

    Denn.5Ad_f?

    FwithsuppOrt[0'..)aidt0bein(subexponentia1Variati.n)

    ifandonlyif,"

    lim

    Z—?..

    ()

    F)

    h

    f

    ol

    F

    dny2(or)equivalently,forsome2),whereFden.testhen-f0ldsc.nv0luti.n

    F?M0regeaFc.ncentrated.n(.., ..)isstil1saidt.bethesubex.

    pv

    .

    ar

    .

    i

    .

    ation

    ifandonlyifF()l<ox<o.,isasubexponentialvariation,where1Adenotesth)

    eindicatortunctionofA. nomBinghametal[1].rEmbrechtsetal4j, wekn.wthat

    CERVCCCCnCS

    T0analyzetheheavytailedclasse8 ,

    forany>0,set

    ?一iminO0}.,,

    O

    u

    0

    p

    ,l:rl

    Then,define,whichiscaliedtheupperMatuszewskainde

    x,as

    +

    =

    inf

    <)logF,(y)

    and,thelowerMatuszewskaindex

    ,as

    =SUp

    <一百logF*(v):

    Themomentindexlieisdefinedby

    ?F:sup{v:.<..)

    .OandF?ifand.nlyifjF+<?.

    weknowthatifF?ERV()OL,))for8oe m

    Vl

    0

    

    ha

    t.H

    kd

    ,

    g

    n

    '=

    T

    t

    o

    e6

    ?3

    aa

    hn

    mm

    No.3Mingetal:UNIFORMESTIMATEONFINITETIMERUINPROBABILITIES693

    and,0..,then,]IF,andifF?74.forsomeOZ,0.., then,;=?F==.AccordingtoProposition2.2.1ofBinghametal[1],weknowthat,

    foranyPl,0<pl<,andP2,<p2<..,therearepositiveconstantsandDi,i=l,2,

    suchthattheinequality (

    holdswheneverYD1,andtheinequality

    F(x()\/

    holdswheneverXYD2.Thus,foranyP>

    

    p=o(F())

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